## Transcribed Text

1. Sketch the parametric curve x = cost, y = cos(2t) (0 < t ≤ 2π), showing its direction.
Find a Cartesian Equation in x and y whose graph contains the parametric curve.
2. Find the coordinates of the points at which the parametric curve x = t
2
, y = t
3 − 3t
has
a) a horizontal tangent
b) a vertical tangent.
3. Find the Cartesian equation of the tangent to the parametric curve x = 3t + t
3
, y =
t
3 − 9t
2 at the point t = 1.
4. For the curve x = t − sin t, y = 1 − cost, nd d
2y
dx2
, and determine where the curve is
concave up and concave down.
5. Find the length of the curve x = t − sin t, y = 1 − cost, 0 ≤ t ≤ 2π. Hint: to
evaluate the integral use the identity sin2
θ =
1−cos(2θ)
2
.
6. Find the area of the surface generated by rotating the curve x = 3t − t
3
, y = 3t
2
, 0 ≤
t ≤ 1 about the x-axis.
7. Transform the polar equation r =
5
3−4 sin θ
to rectangular coordinates.
8. Find the area of the region that lies inside the graphs of both polar equations r = 1
and r = 2 sin θ.
9. Find the length of the polar curve r = e
−θ
, 0 ≤ θ ≤ 3π.
10. Find the limit of the sequence an =
ln(2+e
n)
3n
, if it exists.
11. Find the sum of the series P∞
n=0 3
4
n
5n+1
12. Calculate the sum of the series:
X∞
n=1
1
n(n + 2)
13. Determine whether each of the following series converge or not. (Name the test you
use. You do not have to evaluate the sums of those series.)
a) P∞
n=1
√
n
n3+1
b) P∞
n=2
1
n ln n
c) P∞
n=1
n!
nn .
14. Determine whether each of the following series converge absolutely, converge conditionally or diverge.
1
a) P∞
n=1
(−1)n
n
√
n
b) P∞
n=0(−1)n−1 n
n2+1
15. Starting with the geometric series 1 + x − x
2 + x
3 + . . . (−1 < x < 1), nd
the power series for x
(1−x)
2
in powers of x.
a) Where is the series valid ?
b) Using the result in (a) nd the sum P∞
n=1
n
2n
16. Determine the radius and interval of convergence of the series P∞
n=0
(2x−1)n
3n .
17. Find the Maclaurin series for f(x) = x
2
e
2x
starting with a familiar series. For
that values of x is the representation valid ?
18. Find the Taylor series representation of f(x) = ln x in powers of x − 1. Find the
radius of convergence of this Taylor series.

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